Subgroups — C₃×#33726 #10652333

Order 768.

490 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3160	C₂	1	2	yes	maximal
2	C₃×#2254	C₂	1	2	yes	maximal, complemented
3	C₃×#2253	C₂	1	4	yes	maximal, complemented
4	C₃×#2252	C₂	1	2	yes	maximal, complemented
5	C₃×#2158	C₂	1	4	yes	maximal
6	C₂×C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
7	#33726	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₃×#533	—	2	2	no	—
9	C₂²×C₃×C₁₆	C₂²	1	2	yes	—
10	C₃×#500	C₂²	1	4	yes	—
11	C₃×#499	—	2	2	no	—
12	C₃×#499	C₂²	1	4	yes	—
13	C₃×#498	—	2	2	no	—
14	C₃×#427	—	2	2	no	—
15	C₃×#427	C₂²	1	4	yes	—
16	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#367	C₂²	1	8	yes	—
18	C₃×#366	C₂²	1	8	yes	—
19	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
20	C₃×C₄×C₁₆	C₂²	1	2	yes	—
21	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
22	#3160	C₂×C₃	1	2	yes	—
23	#2254	C₂×C₃	1	2	yes	complemented
24	#2253	C₂×C₃	1	4	yes	complemented
25	#2252	C₂×C₃	1	2	yes	complemented
26	#2158	C₂×C₃	1	4	yes	—
27	C₂×C₄×C₁₆	C₂×C₃	1	1	yes	characteristic
28	C₂²×C₃×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₄⋊Q₈	—	2	4	no	—
31	C₃×#125	—	2	4	no	—
32	C₃×C₂²⋊Q₈	—	4	4	no	—
33	C₃×C₄×Q₈	—	4	4	no	—
34	C₃×#117	—	2	4	no	—
35	C₂×C₃×C₄²	D₄	1	1	yes	characteristic
36	C₂×C₃×C₁₆	Q₈	1	8	yes	complemented
37	C₂×C₃×C₁₆	C₂³	1	2	yes	—
38	C₂×C₃×C₁₆	C₂³	1	2	yes	—
39	C₃×#107	—	2	4	no	—
40	C₃×#107	C₂³	1	8	yes	—
41	C₃×#106	—	2	4	no	—
42	C₃×#106	—	2	4	no	—
43	C₃×C₄×C₈	D₄	1	1	yes	characteristic
44	C₃×C₄×C₈	D₄	1	1	yes	characteristic
45	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
47	#533	—	2	2	no	—
48	C₂²×C₁₆	C₂²×C₃	1	2	yes	—
49	#500	C₂²×C₃	1	4	yes	—
50	#499	—	2	2	no	—
51	#499	C₂²×C₃	1	4	yes	—
52	#498	—	2	2	no	—
53	#427	—	2	2	no	—
54	#427	C₂²×C₃	1	4	yes	—
55	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
56	#367	C₂²×C₃	1	8	yes	—
57	#366	C₂²×C₃	1	8	yes	—
58	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₄×C₁₆	C₂²×C₃	1	2	yes	—
60	C₄×C₁₆	C₂²×C₃	1	1	yes	characteristic
61	C₂×C₃×Q₈	—	4	4	no	—
62	C₂²×C₃×C₄	D₈	1	2	yes	—
63	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₈	Q₁₆	1	4	yes	—
65	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₈	C₂×Q₈	1	4	yes	—
69	C₂×C₃×C₈	C₂⁴	1	1	yes	characteristic
70	C₃×C₄⋊C₄	—	2	8	no	—
71	C₃×C₄⋊C₄	—	4	8	no	—
72	C₃×C₄⋊C₄	—	4	4	no	—
73	C₃×C₂²⋊C₄	—	4	4	no	—
74	C₃×C₄²	—	4	4	no	—
75	C₃×C₄²	D₈	1	2	yes	—
76	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
77	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₃×C₁₆	C₂×Q₈	1	4	yes	—
79	C₃×C₁₆	C₂×Q₈	1	4	yes	—
80	C₂²×C₈	C₃×D₄	1	1	yes	characteristic
81	C₂²×C₈	C₂³×C₃	1	1	yes	characteristic
82	C₄⋊Q₈	—	2	4	no	—
83	#125	—	2	4	no	—
84	C₂²⋊Q₈	—	4	4	no	—
85	C₄×Q₈	—	4	4	no	—
86	#117	—	2	4	no	—
87	C₂×C₄²	C₃×D₄	1	1	yes	characteristic
88	C₂×C₁₆	C₃×Q₈	1	8	yes	complemented
89	C₂×C₁₆	C₂³×C₃	1	2	yes	—
90	C₂×C₁₆	C₂³×C₃	1	2	yes	—
91	#107	—	2	4	no	—
92	#107	C₂³×C₃	1	8	yes	—
93	#106	—	2	4	no	—
94	#106	—	2	4	no	—
95	C₄×C₈	C₃×D₄	1	1	yes	characteristic
96	C₄×C₈	C₃×D₄	1	1	yes	characteristic
97	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
98	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
99	C₂³×C₃	C₂×D₈	1	1	yes	characteristic
100	C₃×Q₈	—	8	8	no	—
101	C₂×C₃×C₄	—	2	4	no	—
102	C₂×C₃×C₄	—	4	8	no	—
103	C₂×C₃×C₄	C₄⋊Q₈	1	2	yes	—
104	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
105	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
106	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
108	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
109	C₃×C₈	C₄⋊Q₈	1	2	yes	—
110	C₃×C₈	C₂×Q₁₆	1	2	yes	—
111	C₃×C₈	C₂×Q₁₆	1	2	yes	—
112	C₃×C₈	C₂²×Q₈	1	2	yes	—
113	C₂×Q₈	—	4	4	no	—
114	C₂²×C₄	C₃×D₈	1	2	yes	—
115	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
116	C₂×C₈	C₃×Q₁₆	1	4	yes	—
117	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
118	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
119	C₂×C₈	C₂×C₃×D₄	1	1	yes	characteristic
120	C₂×C₈	C₂×C₃×Q₈	1	4	yes	—
121	C₂×C₈	C₂⁴×C₃	1	1	yes	characteristic
122	C₄⋊C₄	—	2	8	no	—
123	C₄⋊C₄	—	4	8	no	—
124	C₄⋊C₄	—	4	4	no	—
125	C₂²⋊C₄	—	4	4	no	—
126	C₄²	—	4	4	no	—
127	C₄²	C₃×D₈	1	2	yes	—
128	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
129	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
130	C₁₆	C₂×C₃×Q₈	1	4	yes	—
131	C₁₆	C₂×C₃×Q₈	1	4	yes	—
132	C₂²×C₃	—	2	2	no	—
133	C₂²×C₃	#500	1	2	yes	—
134	C₂²×C₃	C₂²×D₈	1	1	yes	characteristic
135	C₃×C₄	—	2	2	no	—
136	C₃×C₄	—	8	8	no	—
137	C₃×C₄	#500	1	2	yes	—
138	C₃×C₄	#508	1	4	yes	—
139	C₃×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
140	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
141	C₂³	C₂×C₃×D₈	1	1	yes	characteristic
142	Q₈	—	8	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	4	8	no	—
145	C₂×C₄	C₃×C₄⋊Q₈	1	2	yes	—
146	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
147	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
148	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
149	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
150	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
151	C₈	C₃×C₄⋊Q₈	1	2	yes	—
152	C₈	C₂×C₃×Q₁₆	1	2	yes	—
153	C₈	C₂×C₃×Q₁₆	1	2	yes	—
154	C₈	C₂²×C₃×Q₈	1	2	yes	—
155	C₂×C₃	—	2	2	no	—
156	C₂×C₃	C₂×#500	1	1	yes	characteristic
157	C₂×C₃	C₂×#508	1	2	yes	—
158	C₂²	—	2	2	no	—
159	C₂²	C₃×#500	1	2	yes	—
160	C₂²	C₂²×C₃×D₈	1	1	yes	characteristic
161	C₄	—	2	2	no	—
162	C₄	—	8	8	no	—
163	C₄	C₃×#500	1	2	yes	—
164	C₄	C₃×#508	1	4	yes	—
165	C₄	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
166	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
167	C₃	#33726	1	1	yes	characteristic, direct product factor, Sylow
168	C₂	—	2	2	no	—
169	C₂	C₂×C₃×#500	1	1	yes	characteristic
170	C₂	C₂×C₃×#508	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃
socle	C₂²×C₃	C₂²×D₈
prime radical	C₂³×C₃	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃×#2254	C₂	semidirect
3. C₃×#2253	C₂	semidirect
4. C₃×#2252	C₂	semidirect
7. #33726	C₃	direct (N × H)
23. #2254	C₂×C₃	semidirect
24. #2253	C₂×C₃	semidirect
25. #2252	C₂×C₃	semidirect
36. C₂×C₃×C₁₆	Q₈	semidirect
88. C₂×C₁₆	C₃×Q₈	semidirect
167. C₃	#33726	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#3160	2
C₃×#2254	2
C₃×#2253	2
C₃×#2252	2
C₃×#2158	2
C₂×C₃×C₄×C₁₆	2
#33726	3

Subgroups — C3×#33726 #10652333

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#33726 #10652333